Everyday Shea wouldn't be my choice of a base since there is more shea than emulsifiers, nor FIN because there is only one emulsifier/ emollient. Plus I can't see the point in using either as a deep treatment, shea butter is not considered a penetrating oil/ butter nor are many of the oils in the FIN.
FIN may feel drying due to the pH, the lack of slippy emollients, the waxy cetyl esters, I can't say for sure not having used it nor knowing the precise formula. It's not simply oils that make hair feel conditioned in fact oil free condiioners can do the task: the reduction in friction imparted by cationic surfactants and fatty alcohols, flattening the cuticle with an acidic pH also play key roles.
If it were me I'd have a crack at adding more oils to the Body Shop Rainforest moisture since you have it available, but nowhere near as much fatty ingredients much as you did. But why would you want to add even more oils with an extra emulsifier, what exactly are you trying to achieve? It's not deep conditioning if you barely use any conditioning agents relative to the occlusives, you may as well do a straight up oil treatment overnight on dry hair as per the published research, wash the excess out with shampoo or conditioner.
You could purchase and add one of the polysorbates - these are used by some doing the Oil Cleansing Method for skin - but that isn't going to have any particular haircare benefits. By contrast fatty alcohols and cationic surfactants are multi functional - acting as emulsifiers and conditioning agents.